Spectral theorem for compact Hermitian operators

A compact Hermitian operator has real nonzero eigenvalues of finite multiplicity, with zero as their only possible accumulation point, and an orthonormal eigenbasis after a basis of its kernel is included.

For a compact positive self-adjoint operator $T$ with decreasing eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots$, the Courant–Fischer principle givesIn particular, $S\ge T$ in quadratic-form order implies ${\lambda }_k(S)\ge {\lambda }_k(T)$ for every $k$.

Ifthen truncating the sum after $N$ terms gives finite-rank Hermitian $T_N$ with